List of top Multivariable Calculus Questions asked in IIT JAM MA

For a > b > 0, consider
$D=\left\{(x,y,z) \in \R^3 :x^2+y^2+z^2 \le a^2\ \text{and } x^2+y^2 \ge b^2\right\}.$
Then, the surface area of the boundary of the solid D is

Define the function $f: \R^2 → \R$ by
f(x, y) = 12xy e^-(2x+3y-2).
If (a, b) is the point of local maximum of f, then f(a, b) equals

Let
$𝑆 = x\left\{(𝑥, 𝑦) ∈ ℝ^2 : 𝑥 > 0, 𝑦 > 0\right\} ,$
and f: S → ℝ be given by
$f(x,y)=2x^2+3y^2-\log x-\frac{1}{6}\log y.$
Then, which of the following statements is/are TRUE ?

The area of the region
$R=\left\{(x,y) \in \R^2\ : 0\le x \le1,0 \le y \le 1\ \text{and}\ \frac{1}{4} \le xy \le \frac{1}{2} \right\}$
is __________ (rounded off to two decimal places).

The value of
$\lim\limits_{t→\infin}\left((\log(t^2+\frac{1}{t^2}))^{-1} \int\limits_{1}^{\pi t} \frac{\sin^25x}{x}dx\right)$
equals ___________ (rounded off to two decimal places).

Let T be the planar region enclosed by the square with vertices at the points (0,1), (1,0), (0, −1) and (−1,0). Then, the value of
$\iint\limits_T\left(\cos(\pi(x-y))-\cos(\pi(x+y))\right)^2dx\ dy$
equals ____________ (rounded off to two decimal places).

Let
S = {(x, y, z) ∈ $\R^3$ : x² + y² + z² < 1}.
Then, the value of
$\frac{1}{\pi}\iiint_s\left((x-2y+z)^2+(2x-y-z)+(x-y+2z)^2\right)dxdydz$
equals ________ (rounded off to two decimal places).

For n ∈ $\N$, if
$a_n=\frac{1}{n^3+1}+\frac{2^2}{n^3+2}+...+\frac{n^2}{n^3+n}$
then the sequence $\left\{a_n\right\}_{n=1}^{\infin}$ converges to ____________ (rounded off to two decimal places)

Let c > 0 be such that
$\int^c_0e^{s^2}ds=3$
Then, the value of
$$$\int\limits_{0}^c\left(\int\limits^c_0 e^{x^2+y^2}dy\right)dx$
equals __________(rounded off to one decimal place).

Suppose f : (−1, 1) → $\R$ is an infinitely differentiable function such that the series $\sum\limits_{j=0}^{\infin}a_j\frac{x^j}{j^!}$ converges to f(x) for each x ∈ (−1, 1), where,
$a_j=\int\limits_{0}^{\pi/2}\theta^j\cos^j(\tan\theta)d\theta+\int\limits^{\pi}_{\pi/2}(\theta-\pi)^2\cos^j(\tan\theta)d\theta$
for j ≥ 0. Then

Let f(x, y) = $e^{x^2}+y^2$ for (x, y) ∈ $\R^2$ , and a_n be the determinant of the matrix
$\begin{pmatrix} \frac{∂^2f}{∂x^2} &  \frac{∂^2f}{∂x∂y} \\  \frac{∂^2f}{∂y∂x} &  \frac{∂^2f}{∂y^2} \end{pmatrix}$
evaluated at the point (cos(n),sin(n)). Then the limit $\lim\limits_{n \rightarrow \infin}$ a_n is

The area of the curved surface
$S = \left\{ (x, y, z) ∈ \R^3 : z^2 = (x − 1)^2 + (y − 2)^2 \right\}$
lying between the planes z = 2 and z = 3 is

The sum of the series $\sum\limits_{n=1}^{\infin}\frac{2n+1}{(n^2+1)(n^2+2n+2)}$ is equal to _________.(rounded off to two decimal places)

Let f : $\R^2 → \R$ be defined as follows :
$f(x,y)=\begin{cases}    \frac{x^4y^3}{x^6+y^6} & \text{if }(x,y) \ne (0,0)\\     0  & \text{if } (x,y)=(0,0) \end{cases}$
Then

A subset S ⊆ $\R^2$ is said to be bounded if there is an M > 0 such that |x| ≤ M and |y| ≤ M for all (x, y) ∈ S. Which of the following subsets of $\R^2$ is/are bounded ?

Let
$f(x,y)=\iint\limits_{(u-x^2)+(v-y)^2 \le 1}e^{-\sqrt{(u-x)^2+(v-y)^2}}du\ dv.$
Then $\lim\limits_{n \rightarrow \infin}f(n,n^2)$ is

Let f : $\R^2 → \R$ be the function defined as follows :
$f(x,y)=\begin{cases}     (x^2-1)^2\cos^2(\frac{y^2}{(x^2-1)^2}) & \text{if }x \ne ±1 \\     0 & \text{if } x=±1\end{cases}$
The number of points of discontinuity of f(x, y) is equal to _________.

For each t ∈ (0, 1), the surface P_t in $\R^3$ is defined by
$P_t = \left\{(x, y, z) : (x^2 + y^2 )z = 1, t^2 ≤ x^2 + y^2 ≤ 1\right\}.$
Let a_t ∈ R be the surface area of P_t. Then

Let S be the triangular region whose vertices are (0, 0), $(0,\frac{\pi}{2})$ and $(\frac{\pi}{2},0)$. The value of $\iint\limits_S\sin(x)\cos(y)dx\ dy$ is equal to ________. (rounded off to two decimal places)

Let f : $\R^2 → \R^2$ be defined by f(x, y) = (e^x cos(y), e^x sin(y)). Then the number of points in $\R^2$ that do NOT lie in the range of f is

Which of the following functions is/are Riemann integrable on [0, 1] ?

The value of
$\lim\limits_{n\rightarrow \infin}\left(n\int\limits^1_0\frac{x^n}{x+1}dx\right)$
is equal to __________ . (rounded off to two decimal places)

The limit
$\lim\limits_{a\rightarrow0}\left(\frac{\int\limits_{0}^{a}\sin(x^2)dx}{\int\limits_{0}^{a}(\ln(x+1))^2dx}\right)$
is

The value of
$\int^1_0\int_0^{1-x}\cos(x^3+y^2)dy\ dx-\int^1_0\int_0^{1-x}\cos(x^3+y^2)dx\ dy$
is

Let V be the volume of the region S ⊆ $\R^3$ defined by
S = {(x, y, z) ∈ $\R^3$ : xy ≤ z ≤ 4, 0 ≤ x² + y² ≤ 1}.
Then $\frac{V}{π}$ is equal to ________ . (rounded off to two decimal places)